Deficit Solid Angle Research Articles

Modeling the dynamics of global monopoles

A thin wall approximation is exploited to describe a global monopole coupled to gravity. The core is modeled by de Sitter space, its boundary by a thin wall with a constant energy density, and its exterior by the asymptotic Schwarzschild solution with negative gravitational mass $M$ and solid angle deficit, $\ensuremath{\Delta}\ensuremath{\Omega}/4\ensuremath{\pi}=8\ensuremath{\pi}G{\ensuremath{\eta}}^{2}$, where $\ensuremath{\eta}$ is the symmetry-breaking scale. The deficit angle equals $4\ensuremath{\pi}$ when $\ensuremath{\eta}=1/\sqrt{8\ensuremath{\pi}G}\ensuremath{\equiv}{M}_{p}.$ We find that (1) if $\ensuremath{\eta}<{M}_{p},$ there exists a unique globally static nonsingular solution with a well-defined mass, ${M}_{0}<0$. ${M}_{0}$ provides a lower bound on $M$. If ${M}_{0}<M<0$, the solution oscillates. There are no inflating solutions in this symmetry-breaking regime. (2) If $\ensuremath{\eta}>~{M}_{p},$ nonsingular solutions with an inflating core and an asymptotically cosmological exterior will exist for all $M<0$. (3) If $\ensuremath{\eta}$ is not too large, there exists a finite range of values of $M$ where a noninflating monopole will also exist. These solutions appear to be metastable towards inflation. If $M$ is positive, all solutions are singular. We provide a detailed description of the configuration space of the model for each point in the space of parameters $(\ensuremath{\eta},M)$ and trace the wall trajectories on both the interior and the exterior spacetimes. Our results support the proposal that topological defects can undergo inflation.

Quantum gravity with linear action and intrinsic rigidity of spacetime

An earlier proposed theory with linear-gonihedric action A( M 4) for quantum gravity, which requires the existence of a fundamental constant of dimension one is reviewed. One can consider this theory as a “square root” of classical gravity. We also demonstrate that the partition function for the discretized version of the Einstein-Hilbert action found by Regge in 1961 can be represented as a superposition of random surfaces with Euler character as an action, and in the case of linear gravity, as a superposition of three-dimensional manifolds with an action which is proportional to the total solid angle deficit of these manifolds. This representation allows us to construct the transfer matrix which describes the propagation of spatial manifold. We discuss the so called gonihedric principle which allows to define a discrete version of higher derivative terms in quantum gravity and to introduce generalized deficit angles which suppress non-flat vertices, thus introducing an intrinsic rigidity of spacetime. This note is based on a talk delivered at the II meeting on constrained dynamics and quantum gravity at Santa Margherita Ligure.

Quasi-asymptotically flat spacetimes and their ADM mass

We define spacetimes that are asymptotically flat, except for a deficit solid angle and present a definition of their `ADM' mass, which is finite for this class of spacetimes, and, in particular, coincides with the value of the parameter M of the global monopole spacetime studied by Vilenkin and Barriola. Moreover, we show that the definition is coordinate independent, and explain why it can, in some cases, be negative.

WEAK GRAVITATIONAL FIELD AROUND A GLOBAL TEXTURE

Weak gravitational field in the asymptotic region far outside the core of a global texture is investigated without the nonlinear σ-model approximation. We solve the linearized Einstein field equations with the stress-energy tensor contributed by a spherically symmetric global texture to obtain the asymptotic form of the metric. It is shown that, even in this more general case than that of the self-similar solution in the nonlinear σ-model approximation, the metric of the spatial hyperspace in the asymptotic region is that of a flat space with a deficit solid angle whose magnitude is independent of time.

Classical and quantum scattering from global monopoles

The scattering of test particles by a global monopole represented by a spacetime with a solid deficit angle is studied. It is found that the geodesics are equivalent to those of a particle in 2+1 dimensions. For quantum scattering a partial wave analysis is made to obtain analytic expressions for the phase shifts scattering amplitudes, cross sections and the total wavefunction. The results are compared with the analogous study for local cosmic strings and with the S-matrix obtained when both topological defects travel at ultra-relativistic speeds.

GRAVITATIONAL FIELD OF A GAUGE MONOPOLE

It is shown that, in contrast to the global monopoles, no deficit solid angle appears in the solution of the Einstein-Yang-Mills system with Higgs coupling.

General relativistic collapse of textures

We present an exact self-similar solution of the coupled Einstein-σ model equations which describes the general relativistic collapse of global textures. In one coordinate system the texture geometry has a simple interpretation in terms of a deficit solid angle. We also briefly discuss the behavior of matter and light in this geometry. In particular we show that the weak field approximation for the metric perturbations of flat space texture solutions is quantitatively quite reliable.

Gravitational effects of global textures.

A solution for the dynamics of global textures is obtained. Their gravitational field during the collapse and the subsequent evolution is found to be given solely by a space-time dependent deficit solid angle.'' The frequency shift of photons traversing this gravitational field is calculated. The space-time dependent texture metric locally contracts the volume of three-space and thereby induces overdensities in homogeneous matter distributions. There are no gravitational forces unless matter has a nonzero angular momentum with respect to the texture origin which would be the case for moving textures.

Vacuum-polarization effects in global monopole space-times.

The gravitational effect produced by a global monopole may be approximated by a solid deficit angle. As a consequence, the energy-momentum tensor of a quantum field will have a nonzero vacuum expectation value. Here we study this "vacuum-polarization effect" around the monopole. We find explicit expressions for both ${〈{\ensuremath{\varphi}}^{2}〉}_{\mathrm{ren}}$ and ${〈{T}_{\ensuremath{\mu}\ensuremath{\nu}}〉}_{\mathrm{ren}}$ for a massless scalar field. The back reaction of the quantum field on the monopole metric is also investigated.

Repulsive gravitational effects of global monopoles.

A monopole formed as a consequence of the spontaneous breakdown of a global symmetry should have a mass that grows linearly with the distance off its core. It was recently shown by Barriola and Vilenkin that the gravitational effect of this configuration is equivalent to that of a deficit solid angle in the metric, plus that of a relatively tiny mass at the origin. Here we show that this small effective mass is negative. Global monopoles thus share with other topological defects, such as domain walls and global strings, a repulsive gravitational potential. We solve numerically the coupled equations for the metric and the scalar field, to precisely determine this repulsive potential and in order to analyze the solution when gravitational effects are already significant close to the monopole core. We study the motion of test particles in a monopole background, and discuss the possible implications of a negative effective mass.

Gravitational field of a global monopole.

We present an approximate solution of the Einstein equations for the metric outside a monopole resulting from the breaking of a global O(3) symmetry. The monopole exerts practically no gravitational force on nonrelativistic matter, but the space around it has a deficit solid angle, and all light rays are deflected by the same angle, independent of the impact parameter.